On optimizing over lift-and-project closures

TL;DR

This paper introduces a simple LP-based algorithm to approximate the lift-and-project closure in mixed integer linear programming, connecting it to disjunctive programming and Gomory cuts, with computational validation.

Contribution

It proposes a novel, straightforward LP formulation for approximating the lift-and-project closure, linking it to existing cutting-plane methods and providing computational evidence.

Findings

The proposed LP method effectively approximates the lift-and-project closure.

The separation LP relates to disjunctive programming and Gomory cuts.

Computational experiments show competitive performance.

Abstract

The lift-and-project closure is the relaxation obtained by computing all lift-and-project cuts from the initial formulation of a mixed integer linear program or equivalently by computing all mixed integer Gomory cuts read from all tableau's corresponding to feasible and infeasible bases. In this paper, we present an algorithm for approximating the value of the lift-and-project closure. The originality of our method is that it is based on a very simple cut generation linear programming problem which is obtained from the original linear relaxation by simply modifying the bounds on the variables and constraints. This separation LP can also be seen as the dual of the cut generation LP used in disjunctive programming procedures with a particular normalization. We study some properties of this separation LP in particular relating it to the equivalence between lift-and-project cuts and Gomory cuts shown by Balas and Perregaard. Finally, we present some computational experiments and comparisons with recent related works.